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 [% 2s] Electron Density in 2s/2p Hybrid Orbital
Color shows the sign of the orbital, which has been squared to give density.
The tiny white cross marks the position of the nucleus.

(Don't let the graphic appeal of this animation keep you from
thinking about the mathematical formula for the hybrid!)

Hybridization of an Atom in an Electric Field

If an hydrogen atom were placed in an electric field that pulled the electron toward the left and the proton toward the right, there would be no net force on the atom, so that the center of mass (the proton, for practical purposes) would stay in place, but the electron cloud would shift toward the left and distort away from spherical symmetry.

If the coulombic field of the nucleus is much stronger than the "perturbing" field, the proper low-energy solutions of the Schrödinger equation should not change too much from those appropriate for the atom alone. A reasonable guess for the proper atomic orbital in the presence of the perturbing field is thus a mixture of the low-energy atomic orbitals that are observed in the absence of the field. Such mixtures are called hybrid orbitals. They are generated simply by adding the functions in various proportions.

The field-free orbital of lowest energy is 1s. Adding a little 2px to the 1s orbital would give an orbital with larger numbers on one side of the nucleus and smaller numbers on the other. [In fact adding 2.7% of 2px to the 1s orbital improves the wave function for the H2 molecule - click here for illustration.]

It is costly in terms of energy to mix the 2p orbital with the 1s, because the 2p function is much more energetic (stronger curvature/amplitude). It is easier to mix 2p with 2s, because they have similar energies (of course the resulting hybrid orbital would have a still higher energy than a 1s/2p hybrid, but the mixing itself is easier). This is why the valence orbitals of carbon hybridize more extensively than those of hydrogen.

The pictures above were generated by Dean Dauger's "Atom in a Box" program (play with the real thing to have more fun). They show how electron density shifts when the 2px orbital is mixed with 2s, as in carbon valence hybrids. The number at the left of each view indicates the % of 2s orbital density in the hybrid, i.e. the square of a in the formula (a 2s + b 2px).

Note that a very modest amount of 2px character (like 2%) already causes a substantial shift, and that the shift maximizes at about 50% 2px character. The distribution becomes symmetrical again in the pure 2px orbital.

As a tiny perturbing field is applied to a "ground state" 1s H atom and then ramped up, the atomic orbital would change from 1s with negligible field to something like sp with very high field. [An even higher field could shift the electron cloud further by beginning to mix in orbitals of higher energy.]

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Go to MOs in QMech for Chem 125
Overlap of various C Hybrid Orbitals

copyright 2001-2003 J.M.McBride